Constrained High-Index Saddle Dynamics for the Solution Landscape with Equality Constraints

• Main Authors: Huang, Z. (Author), Yin, J. (Author), Zhang, L. (Author)
• Format: Article
• Language: English
• Published: Springer 2022
• Subjects: Bose-Einstein condensation; Bose–Einstein condensation; Bose-Einstein-condensation; Condensation; Constrained optimization; Energy landscape; Equality constraints; Higher index; Manifold optimization; Numerical methods; Optimisations; Saddle dynamics; Saddle point; Solution landscape; Thomson problem
• Online Access: View Fulltext in Publisher

 We propose a constrained high-index saddle dynamics (CHiSD) method to search for index-k saddle points of an energy functional subject to equality constraints. With Riemannian manifold tools, the CHiSD is derived in a minimax framework, and its linear stability at an index-k saddle point is proved. To ensure the manifold property, the CHiSD is numerically implemented using retractions and vector transport. Then we present a numerical approach by combining CHiSD with downward and upward search algorithms to construct the solution landscape in the presence of equality constraints. We apply the Thomson problem and the Bose–Einstein condensation as numerical examples to demonstrate the efficiency of the proposed method. © 2022, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.